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• Please show your work

• Please give focused answers to the question

EC opportunity – Solar System Planetarium Show, Jan 27th, sign up now

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Retrograde motion

a) is explained by the geocentric model as planets moving backwards sometimes

b) is explained by the heliocentric models as planets passing each other.

c) is seen for the planets, but not the Sun and Moon

d) was studied in ancient Greece

e) All the other answers are correct

f) Copernican system did.

Lecture 5: Lecture 5: Kepler & the Harmony of the PlanetsKepler & the Harmony of the Planets

Astronomy 1143 – Spring 2014Astronomy 1143 – Spring 2014

Key IdeasKepler – mathematician w/Tycho’s data

Kepler described planetary orbits by 3 Laws

• 1st Law: planets move in ellipses with the Sun at one focus

• 2nd Law: line connecting Sun and planets sweeps out equal areas in equal time

• 3rd Law: The square of a planet’s orbital period is proportional to the cube of the semimajor axis of the orbit.

Key Ideas

Important Consequences:• Excellent predictions for positions of planets

• Fit Tycho’s data• Heliocentric model

• Measured ratios of distances in the solar system• Measure one distance = get them all

• Showed what gravity does (Newton)• Take these expectations when observing

galaxies

After Copernicus

Copernicus died in 1543, the year De Revolutionibus was published

Astronomers became familiar with his work for its mathematics, even if they did not accept his cosmology

But certain astronomers who read his work thought that heliocentrism was correct

• Galileo

• Kepler

Johannes Kepler (1571-1630)Brilliant German Mathematician

• Staunch Copernican• Convinced the Universe was

governed by physical laws.• Obsessed with finding

harmony in the heavens.• Had a genius for data analysis

Inherited Tycho’s data & post as the Imperial Mathematicus in 1601.

The Motions of MarsMars was the key to unlocking the secrets of planetary motion.

Kepler began analyzing the orbit of Mars in 1601. It took him 4 years:

• Started by determining the orbit of the Earth using successive oppositions of Mars.

• Fit an off-center circle (i.e., not centered on the Sun) to the first 4 data points.

A 5th test data point did not fit by 8 arcminutes

Kepler's calculations of the orbits of the Earth & Mars, from the Astronomia Nova (1609)

Kepler listened to the data:• Knew Tycho’s data were accurate to 12

arcminutes.• But never off by as much as 8 arcminutes.

Kepler questioned his assumptions:• Forced to abandon uniform circular motion.• Concluded Mars’ orbit was not a circle, but

instead an ellipse with the Sun at one focus.

Published results in 1609 (Astronomia Nova).

The 1st Law of Planetary Motion

The orbits of the planets are ellipses with the Sun at one focus.

Ellipses are characterized by two numbers:• Semimajor Axis (a): ½ of length of the

longest axis• Eccentricity (e): shape of the ellipse.

Orbit of Mars:• a = 1.5237 AU, e = 0.0934

F2F1

Semimajor Axis

Center

Focii

The 2nd Law of Planetary Motion

The line joining the Sun and the planet sweeps out equal areas in equal times.

• Planets move fastest at Perihelion

• Planets move slowest at Aphelion

Kepler’s Second Law provides a geometric description of the change in speed.

Completely eliminates epicycles.

T=0d

T=10d

T=0d

T=10d Equal Areas in Equal Times

Let’s go to the animation….

The 3rd Law of Planetary Motion

The square of a planet’s orbital period is proportional to the cube of the semimajor axis of the orbit.

Expressed Mathematically:

2 3P aP = Period in years, a = Semimajor axis in AU

The Astronomical Unit

Distance between Earth and Sun

Length of the semi-major axis of Earth’s orbit

Number of kilometers = ?

The Third Law applies to all bodies orbiting the Sun: planets, comets, rocks, & spacecraft!

Planet a (AU) P (yr) a3 P2

Mercury 0.387 0.241 0.0580 0.0580

Venus 0.723 0.615 0.3785 0.3785

Earth 1.000 1.000 1.0000 1.0000

Mars 1.524 1.881 3.5373 3.5376

Jupiter 5.203 11.863 140.88 140.72

Saturn 9.537 29.447 867.45 867.16

Empirical Laws

Kepler’s Laws are Empirical Laws:• They describe how the planets move.• They don’t explain why they move that way.

Not yet Physical Laws:• Kepler made a start, but he had incorrect

ideas about forces.• Kepler’s thinking was strongly motivated by

his notions of universal harmony.Correct explanation had to wait until the work of Isaac Newton.

Kepler’s Law’s at Work!

Heliocentrism and Kepler

Kepler’s belief that the Sun was the driving force in the Solar System was one of the reasons that he could figure out his laws.

His laws are very Sun-centric.

For example, 2nd Law arose because he thought that the Sun’s rays pushed along the planets in their orbits – farther away = fewer rays = slower speeds

Important Consequences

This is not the last we will hear about Kepler in this course

• Accuracy of predictions of positions of planets important for convincing people that heliocentrism was correct

• Third Law gave the ratios of the distances of planets from Sun – know one distance, know them all

• Laws apply to systems other than our own – except when they unexpectedly don’t…

Rudolphine Tables

In 1627, Kepler published the Rudolphine Tables, based on his theory of how the planets moved around the Sun.

Predicted positions clearly better than tables based on the models of Ptolemy or Copernicus

Definitely noticed by astronomers & part of the reason for the gradual switch to heliocentrism

Rudolphine Tables

Kepler’s Third Law

Speeds in the Solar System

Kepler and Dark Matter

As Kepler showed (and Newton explained), the speeds of the planets are lower for the most distant ones

• Diminishing force of gravity as distance from the largest mass increases

• We expect that should also be the case for stars orbiting the distant edges of galaxies

• But……

Spoiler Alert: Non-Keplerian speeds ahead!

Spoiler Alert: Non-Keplerian speeds ahead!

Spoiler Alert: Non-Keplerian speeds ahead!